Threshold solutions for the focusing generalized Hartree equations

TL;DR

This paper classifies the long-term behavior of solutions to the focusing generalized Hartree equation at the critical mass-energy threshold, identifying special solutions and their stability properties.

Contribution

It introduces a classification of solutions at the threshold, including new special solutions and their asymptotic behaviors, extending previous results below the threshold.

Findings

Identification of three special solutions: $e^{it} Q$, $Q^ ext{+}$, $Q^ ext{−}$.

Classification of all solutions at the threshold as either similar to these solutions, scattering, or blowing up.

Reliance on the uniqueness and non-degeneracy of the ground state.

Abstract

We study the global behavior of solutions to the focusing generalized Hartree equation with $H^1$ data at mass-energy threshold in the inter-range case. In the earlier works of Arora-Roudenko [Arora-Roudenko 2021], the behavior of solutions below the mass-energy threshold was classified. In this paper, we first exhibit three special solutions: $e^{it} Q$, $Q^\pm$, where $Q^\pm$ exponentially approach to the $e^{it} Q$ in the positive time direction, $Q^+$ blows up and $Q^-$ scatters in the negative time direction. Then we classify solutions at this threshold, showing that they behave exactly as the above three special solutions up to symmetries, or scatter or blow up in both time directions. The argument relies on the uniqueness and non-degeneracy of ground state, which we regard as an assumption for the general case.